Determine event independence using P(A and B)=P(A)P(B).
Find a missing probability using independence and \(P(A and B) = P(A)P(B)\)
Problem
For independent \(A\) and \(B\) with \(P(A\cap{}B)=0.12\) and \(P(A)=0.3\), solve \(0.12=0.3P(B)\) for \(P(B)\).
Big Picture
What this problem is really about
Independence turns the joint probability into a product equation with one missing factor. We’ll substitute the known joint and marginal values, divide by the nonzero coefficient to isolate the unknown probability, multiply back to verify, and check that the result lies between zero and one.
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